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\centerline{Mathematics 243, section 1 -- Algebraic Structures}
\centerline{Practice Questions for Exam 1}
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\noindent
Note:  This list is slightly longer than the actual exam will be to 
show the range of different kinds of questions that might appear.  
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I.   Let 
$$\eqalign{U &= {\bf Z}\cr
           A &= \{ x \in {\bf Z} : x {\rm\ is\ even\ and\ } -3 \le x \le 10\}\cr
           B &= \{1,2,3,4,5\}\cr
           C &= \{x \in {\bf Z} : x < -4 {\rm\ or\ } x > 7\}\cr}$$
\item{A)}  What is the set $A \cap (B \cup C)$.
\item{B)}  What is the set $C'$?
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\noindent
II.  Let $A, B$ be any two subsets of a universal set $U$.
Prove that $(A \cap B)' = A' \cup B'$ (where $X'$ is the complement of $X$). 
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III.  Consider the following statement, where $A$ is a nonempty set.
``If $f : A \to B$ is an onto mapping, then for all $b\in B$, the
set $f^{-1}(\{b\})$ has at least one element.''
\item{A)} Give the contrapositive form of this statement.
\item{B)} Give the converse of the statement.  
Is the converse true or false?  Explain.
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IV.  
Let $f,g : {\bf Z} \to {\bf Z}$ be the mappings
$$f = \cases{x+1 & if $x$ is even\cr
             x-1 & if $x$ is odd\cr}
\qquad  g = \cases{x-1 & if $x$ is even\cr
            (x-1)/2 & if $x$ is odd\cr}$$
\item{A)} Let $S = \{1,2,3\}$ and 
$T = \{0,3,6\}$.  What is $g^{-1}(T) \cup f(S)$?
\item{B)}  What is the mapping $f\circ g$? 
\item{C)}  Prove that the mapping $f$ 
is a permutation of ${\bf Z}$.
\item{D)}  Is the mapping $g$ a permutation of ${\bf Z}$?
Prove your assertion.
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\noindent         
V.  Let $*$ be the binary operation on ${\bf R}$
defined by $x*y = 3 x y + x + y$. 
\item{A)} Prove that 
$*$ is commutative. 
\item{B)} Is $*$ associative? Prove your assertion.
\item{C)} Does $*$ 
have an identity element? Prove your assertion.
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\noindent
VI.  
\item{A)}  Let $M = M_{m\times n}({\bf R})$ be the set of 
all $m\times n$ matrices with real entries.  Show that
the operation of matrix addition is a commutative and associative
binary operation on $M$.
\item{B)}  Let $O_{n\times p}$ be the $n\times p$ matrix all
of whose entries are zero.  Show that if $A$ is any 
$m\times n$ matrix, then $AO_{n\times p} = O_{m\times p}$
(the $m\times p$ zero matrix).  
\bye
 
